   Chapter 15, Problem 9RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The integral ∫ 0 2 π ∫ 0 2 ∫ r 2 d z   d r   d θ represents the volume enclosed by the cone z   =   x 2   +   y 2 and the plane z = 2.

To determine

Whether the statement, “The integral 02π02r2dzdrdθ represents volume enclosed by the cone z=x2+y2 and the plane z=2 ” is true or false.

Explanation

Formula used:

The volume of the solid over the given region is given by, V=Edzdxdy .

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π , then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

Reason:

Convert the given problem into cylindrical coordinates. For that, substitute x=rcosθ,y=rsinθ and z=z

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